#
Topics in Algebraic Geometry

# Math 647, Spring 2013

### Course description:

The main prerequisite
is a one year introductory course on algebraic geometry (e.g. chapters II-IV in
Hartshorne's textbook).

The aim of the course
is to study more advanced topics, and this will be organized as follows. The
main goal is to study moduli spaces of smooth curves and their
compactifications, for this we will have to study the following topics: stable
curves, Hilbert schemes, Grothendieck topologies, algebraic spaces and
stacks, moduli spaces of stable
n-pointed curves, and stable reduction theorem. As the main application of the
developed theory we will prove in the end of the course de Jong's theorems on
semi-stable families and desingularization of varieties by alterations.

There is no final
exam. Instead of this, students that want to take the course for credit will
give a talk in the end of the course.

### Texts:

DM, the article by Deligne-Mumford on
irreducibility of moduli spaces.

AO, lecture notes by Abramovich-Oort on de
Jong’s results and related stuff.

Lecture notes, where I wrote
down lectures of an analogous course given at University of Pennsylvania. (You
can ignore the division to lectures since our 3-hour meetings will cover more
material.) Any remark on a mistake/inaccuracy will be very welcome (and I’m
sure that there are some).

###